Operation of a titanium nitride superconducting microresonator detector
in the nonlinear regime
L. J. Swenson,1,2,a)P. K. Day,2B. H. Eom,1H. G. Leduc,2N. Llombart,3C. M. McKenney,1 O. Noroozian,4and J. Zmuidzinas1,2
1
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
3
Delft University of Technology, 2628 CD Delft, The Netherlands
4
Quantum Sensors Group, National Institute of Standards and Technology, Boulder, Colorado 80305, USA (Received 26 November 2012; accepted 22 February 2013; published online 8 March 2013) If driven sufficiently strongly, superconducting microresonators exhibit nonlinear behavior including response bifurcation. This behavior can arise from a variety of physical mechanisms including heating effects, grain boundaries or weak links, vortex penetration, or through the intrinsic nonlinearity of the kinetic inductance. Although microresonators used for photon detection are usually driven fairly hard in order to optimize their sensitivity, most experiments to date have not explored detector performance beyond the onset of bifurcation. Here, we present measurements of a lumped-element superconducting microresonator designed for use as a far-infrared detector and operated deep into the nonlinear regime. The 1 GHz resonator was fabricated from a 22 nm thick titanium nitride film with a critical temperature of 2 K and a normal-state resistivity of 100 lX cm. We measured the response of the device when illuminated with 6.4 pW optical loading using microwave readout powers that ranged from the low-power, linear regime to 18 dB beyond the onset of bifurcation. Over this entire range, the nonlinear behavior is well described by a nonlinear kinetic inductance. The best noise-equivalent power of 2 1016W=Hz1=2at 10 Hz was measured at the highest readout power, and represents a10 fold
improvement compared with operating below the onset of bifurcation.VC 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4794808]
I. INTRODUCTION
Superconducting detector arrays have a wide range of applications in physics and astrophysics.1,2Although the de-velopment of multiplexed readouts has allowed array sizes to grow rapidly over the past decade, there is a strong demand for even larger arrays. For example, current astronomical sub-millimeter cameras feature arrays with up to 104pixels.3In comparison, the proposed Cerro Chajnantor Atacama Telescope (CCAT) 25-meter submillimeter telescope4would require more than 106pixels in order to fully sample the focal plane at a wavelength of k¼ 350 lm. Another example is cryogenic dark matter searches which currently implement arrays with <50 pixels. Each pixel features an integrated transition-edge sensor (TES) to detect athermal phonons for 100–500 g of cryogenic detection mass.5,6 Proposed ton-scale searches would require of order 104detectors. In order to meet these challenging scaling requirements, it is highly de-sirable to simplify detector fabrication and to increase multi-plexing factors in order to reduce system cost. From this perspective, superconducting microresonator detectors7–12are particularly attractive. In these devices, the energy to be detected is coupled to a superconducting film, causing Cooper pairs to be broken into individual electrons or quasiparticles, which leads to a perturbation of the complex ac conductivity drðxÞ ¼ dr1 jdr2. Very sensitive measurements of drðxÞ
may be made if the film is patterned to form a microwave reso-nant circuit. Because both the amplitude and phase of the com-plex transmission of the circuit can be measured (see Fig. 1), information on both the dissipative (dr1) and reactive (dr2)
perturbations may be obtained simultaneously, giving the user a choice of using reactive readout, dissipation readout, or both.
FIG. 1. Measurement setup. (a) Micrograph of a single LEKID-type micro-resonator detector from a 16 16 array. The interdigitated capacitor is visi-ble above the spiral inductor/absorber. The array was fabricated from a TiN film with a critical temperature ofTc¼ 2 K deposited on a high-resistivity
silicon substrate. The 22 nm film thickness was significantly thinner than the effective penetration depth k¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihqn=ðl0pD0Þ
p
750 nm. (b) Electrical and optical setup. The array is refrigerated to 110 mK and read out using a voltage-controlled microwave signal generator and standard homodyne detection electronics. A variable-temperature blackbody is located behind a 4.2 K band-defining 215 lm metal-mesh filter (BW¼ 43 lm). A 7 mm diam-eter aperture on the still shield is further equipped with 1.6 mm of high-density polyethylene and a 300 cm1 low-pass filter. Various electrical signal-conditioning amplifiers and attenuators, including two 20 dB attenua-tors located on the 4.2 K and still stages, are not shown.
a)Electronic mail: swenson@astro.caltech.edu.
Frequency multiplexing of a detector array is readily accom-plished by designing each microresonator to have a different resonant frequency and coupling all of the detectors to a single transmission line for excitation and readout.
Although various options exist for coupling pair-breaking photons or phonons into the superconductor, the simplest approach for a number of applications is to directly illuminate the microresonator. For good performance, the resonator must be designed to be an efficient absorber. A particularly notable example is the structure introduced by Doyleet al. for far-infrared detection, known as the lumped-element kinetic inductance detector or LEKID.13,14Fig.1(a)
shows a variant of this concept, designed for low interpixel crosstalk and polarization-insensitive operation.15The struc-ture consists of a coplanar stripline spiral inductor and an interdigitated capacitor. The microwave current density in the inductor is considerably larger than in the capacitor, so the inductor is the photosensitive portion of the device. The resonant frequency can be tuned simply by varying the ge-ometry of the capacitor or inductor during the array design. Further, only a single lithography step is necessary for patter-ing an array of resonators from a superconductpatter-ing thin-film deposited on an insulating substrate. The simplicity of these devices has led to the demonstration of prototype arrays suit-able for submillimeter astronomy.16 Similar devices have been developed for optical astronomy17 and dark matter detection experiments.18,19
Because the pixel size is comparable to the far-infrared wavelength, the details of the resonator geometry do not strongly affect the absorption of radiation. However, in order to achieve high absorption efficiency, the effective far-infrared surface resistance of the structure should be around Reff¼ 377 X=ð1 þ ffiffiffiffir
p
Þ 86 X when using a silicon sub-strate with dielectric constant r 11:5. This results in an
approximate constraint on the sheet resistance Rs and area filling factor gA of the superconducting film,Reff Rs=gA,
which is straightforward to satisfy if a high-resistivity super-conductor such as TiN is used.20 These considerations provide a starting point for pixel design; detailed electromag-netic simulations may then be used to optimize the absorp-tion. TiN is a particularly suitable material for resonator detectors due to its high intrinsic quality Qi which can exceed 106 and a tunable Tc based on the nitrogen content (0 <Tc< 4:7 K).
In practice, superconducting microresonators exhibit excess frequency noise.7,8,11,21 This noise is due to capaci-tance fluctuations22 caused by two-level tunneling systems that are known to be present in amorphous dielectrics.23,24 Such material is clearly present when deposited dielectric films are used in the resonator capacitor.25However, experi-ments have shown that even when the capacitor consists of a patterned superconducting film on a high-quality crystalline dielectric substrate, a thin surface layer of amorphous dielec-tric material is still present and causes excess dissipation and noise.26,27This two-level system (TLS) noise has been stud-ied extensively and a number of techniques have been devel-oped to reduce it.22,28,29One of the simplest ways to mitigate the effects of TLS noise and simultaneously overcome am-plifier noise is to drive the resonator with the largest readout
power possible.21This technique is ultimately limited by the nonlinear response of the resonator. Potential sources of non-linearity in thin-film superconducting resonators include a power-dependent current distribution;30 quasiparticle pro-duction from absorption of readout photons;31or the nonlin-ear kinetic inductance intrinsic to superconductivity.32–34
Virtually all measurements of microresonator detectors reported to date have used a readout power below the onset of bifurcation. Here, we demonstrate operation of a lumped-element microresonator detector both in the low-power, linear regime, and deep in the nonlinear regime well above the onset of bifurcation. For most of our measurements, the pixel was illuminated with a substantial optical load of Popt¼ 6:4 pW. For comparison, at the highest achievable
readout power (discussed below) the readout power dissi-pated in the resonator was1.6 pW. While this is compara-ble to the optical loading, the efficiency for conversion of this power into quasiparticles is expected and observed35 to be low since the energy of each readout photon is a factor of D=hfr¼ 1:76kBTc=hfr ¼ 73 below the superconducting gap
energy. Much of the dissipated microwave power may be expected to escape as low-energy, non-pair-breaking pho-nons, in which case the quasiparticle population may not change substantially due to the microwave dissipation. As a result, it is perhaps not entirely surprising that the behavior of our device even deep into the bifurcation regime is well described by a model that includes only the nonlinearity of the kinetic inductance.
II. THEORETICAL MODEL AND RESONANCE FITTING
The basic principles of superconducting microresonator detector readout when operating in the linear regime have been extensively described.7,8The homodyne readout used for this measurement is shown in Fig.1(b). A microresonator with an intrinsic, unloaded quality factor Qiand, resonance frequency xr ¼ 1=
ffiffiffiffiffiffi LC p
is coupled to a transmission line, yielding a coupling quality factorQc. A fraction a of the total inductanceL is contributed by the kinetic inductance Lksuch that Lk¼ aL. The overall loaded quality factor is given by
Q1r ¼ Q1i þ Q1c . A signal generator is used to drive the
resonator near its resonance frequency. The transmitted sig-nal is amplified by a cryogenic amplifier with noise tempera-ture Tn¼ 6 K, mixed with a copy of the original signal and digitized. The resulting complex amplitude of the measured signal is described by the forward transfer function
S21¼ 1 Qr Qc 1 1þ 2jQrx ; (1) where x¼xg xr xr (2)
is the fractional detuning of the readout generator frequency xg relative to the resonance frequency xr. Varying x by
sweeping the generator frequency traces out a circle in the complexS21plane. At resonance (xg¼ xr; x¼ 0), the circle
while the S21 values for all generator frequencies far from resonance fall on the real axis near unity.
Increasing, the readout power results in the onset of non-linear behavior. As discussed, the most relevant source of nonlinearity for this device is the nonlinear kinetic induct-ance of the superconducting film. A power-dependent kinetic inductance can be written in terms of the resonator currentI with the expression
LkðIÞ ¼ Lkð0Þ½1 þ I2=I2þ …; (3)
where odd terms are excluded due to symmetry considera-tions andIsets the scaling of the effect.Lkð0Þ is the kinetic
inductance of the resonator in the low-power and linear limit.
The nonlinear kinetic inductance gives rise to classic soft-spring Duffing oscillator dynamics.36In order to quanti-tatively account for the power-dependent behavior, it is nec-essary to replace Eq.(1)with a transfer function which takes into account the resonance shift dxrdue to the nonlinear
ki-netic inductance in Eq.(3). The shifted resonance is given by xr ¼ xr;0þ dxr, where xr;0is the low-power resonance
fre-quency. Substituting into Eq. (2), the generator detuning becomes
x¼xg xr;0 dxr xr;0 dxr
x0 dx; (4)
where the approximation is calculated to first order and x0¼
xg xr;0
xr;0
(5)
is the detuning in the low-power and linear limit. At a stored resonator energy E, the nonlinear frequency shift dx is given by dx¼dxr xr;0 ¼ 1 2 dL L ¼ a 2 I2 I2 ¼ E E ; (6)
where the scaling energyE/ LkI2=a2 is expected to be of
order the condensation energy of the inductor if a 1. To proceed further, an expression for the stored resona-tor energy at a given readout power and frequency is required. The available generator powerPgcan be reflected back to the generator, transmitted past the resonator, or dissi-pated in the resonator. Conservation of power can be expressed by
Pdiss ¼ Pg½1 jS11j2 jS21j2; (7)
wherePdissis the power dissipated in the resonator andS11is the normalized amplitude of the reflected wave. Noting that S11¼ S21 1 for a shunt-coupled circuit and substituting
Eq.(1)into Eq.(7)yields the result
Pdiss¼ Pg 2Q2 r QiQc 1 1þ 4Q2 rx2 : (8)
Using the standard definition of the internal quality factor
Qi¼
xrE
Pdiss
; (9)
the resonator energy is found to be E¼2Q 2 r Qc 1 1þ 4Q2 rx2 Pg xr : (10)
Equation (4) is an implicit equation for the power-shifted detuning x as a function of the generator power Pg and detuning at low power,x0. To see this, recall from Eqs.
(4)and(6)thatx¼ x0þ E=E. Combining this with Eq.(10)
yields x¼ x0þ 2Q2 r Qc 1 1þ 4Q2 rx2 Pg xrE : (11)
Introducing, the variablesy¼ Qrx and y0 ¼ Qrx0 as well as
the nonlinearity parameter a¼2Q 3 r Qc Pg xrE (12)
allows Eq.(11)to be rewritten as y¼ y0þ
a
1þ 4y2: (13)
Using the definition of the quality factor Qr¼ xr=Dx,
where Dx is the linewidth of the resonance, we see that y¼ Qrx¼ ðxg xrÞ=Dx. Thus, y and y0are the generator detuning measured in linewidths relative to the power-shifted resonance and the low-power resonance, respectively. Solutions to Eq.(13)for a range ofa are shown in Fig.2. As can be seen from this plot,y becomes nonmonotonic with y0 fora > 4pffiffiffi3=9 0:8.
FIG. 2. Solutions to Eq.(13)for a range of the nonlinear parametera. The solid (dashed) arrows indicate downward (upward) frequency sweeping. The horizontal scaley0is the generator detuning measured relative to the
low-power resonance frequency xr;0. The vertical scale is the generator detuning
y measured relative to the shifted resonance frequency xr. Fora > 4
ffiffiffi 3 p
=9 0:8, y is nonmonotonic in y0.
The origin of the bifurcation is conceptually simple to understand and is visualized in Fig. 3. If the generator fre-quency is swept upwards starting from below the resonance, the resonator current increases as the detuning decreases, and the nonlinear inductance causes the resonance to shift downward toward the generator frequency, reducing the detuning further. This process eventually results in a run-away positive feedback condition as the resonator “snaps” into the energized state. In contrast, sweeping downwards from the high-frequency side results in negative feedback as the resonance also shifts downwards, away from the genera-tor tone. The generagenera-tor tone chases the resonance downward until sweeping past the resonance minimum when the reso-nator abruptly snaps back to its non-energized state. The maximum frequency shift during downward frequency sweeping will depend on the readout power and reflects the I2 dependence of the kinetic inductance in Eq. (3). Notice that smooth downward frequency sweeping allows access to the entire high-frequency side of the resonance.
Fitting a measured resonance curve yields valuable information including the resonance frequency and quality factors. A fit to Eq.(1) of a calibrated resonance in the low-power, linear regime under 6.4 pW of optical loading is shown in Fig. 4(a). From this fit, we find Qi¼ 8:7 105;
Qc¼ 8:1 105, and the low-power resonance frequency is
fr;0¼ 1:06 GHz. A blind application of Eq.(1)in the
nonlin-ear regime results in a poor fit to the resonance, as exhibited
in Fig. 4(b). Instead, the frequency shifted detuningx at the appropriatePgmust be found from Eq.(11) and substituted into Eq. (1). The nonlinear energy scale E can be
deter-mined from Eq. (12) by carefully measuring the generator power at the onset of bifurcation (a :8). The results both below and well above bifurcation can be seen in Figs. 4(c)
and4(d). Here, the calibration parameters and the low-power fitted quality factors have been fixed. Only the frequency shifted detuningx has been substituted into Eq.(1)yielding good agreement with the measured data over a broad range of generator powers.
The maximum achievable readout power in this device was limited by the abrupt onset of additional dissipation in the resonator. Switching occurred at 18 dB above the bifurca-tion power in this device at a dissipated power of Pdiss
> 1:6 pW. In the S21plane, the new state traces out a circle with a smaller diameter than the original resonance circle. While the source of this additional dissipation is currently under investigation, we can speculate that at a sufficiently high readout photon density in the resonator, multi-photon absorption by the quasiparticles can result in emission of phonons with energy h > 2D. These high energy pho-nons can subsequently break Cooper pairs resulting in an increased quasiparticle density.37 All measurements pre-sented here were taken below the emergence of this behav-ior. Comparing Figs. 4(a) and 4(d), it is evident that the depth of the transfer function on resonance remained con-stant from the linear regime to deep within the bifurcation re-gime. This indicates that the device dissipation is readout power independent. Thus, before the emergence of an addi-tional device state, the nonlinear effects can be completely understood as being reactive and not dissipative in nature.
FIG. 3. Response bifurcation due to feedback. Due to the nonlinear kinetic inductance, the resonator current induced by the generator shifts the reso-nance to lower frequency. (a) Upward frequency sweeping. As the tone enters the resonance (position 2), runaway positive feedback causes the reso-nance to quickly snap to lower frequency (position 3). (b) Downward fre-quency sweeping. As the generator frefre-quency decreases, negative feedback pushes the resonance toward lower frequencies, away from the generator. When the generator goes past the resonance minimum (position 5), the reso-nance snaps back to its unperturbed state (position 1). In both subplots, verti-cal lines indicate various generator frequencies during a frequency sweep. The intersection of a given measurement tone and the corresponding shifted resonance is marked with an X. The locus of these intersections traces out a hysteretic transfer function (dashed lines) as the frequency is swept upwards or downwards.
FIG. 4. Fitting measured resonances for a range of the nonlinear parameter a. In the linear case (a), application of Eq.(1)yields the desired resonance parameters. As the readout power is increased (b), direct use of Eq.(1)is no longer sufficient and results in poor agreement with the data. Instead, the fre-quency shifted detuningx can be calculated from Eq.(11)and substituted into Eq.(1). This approach results in good agreement to the data both below (c) and above (d) the onset of bifurcation.
The condensation energy of the inductor is given by Econd¼ N0D2VL=2, where N0 is the single spin density of states at the Fermi energy, D 3:5kBTc=2 is the
supercon-ducting gap, andVL is the volume of the inductor.38 Econd
¼ 3 1013J for this device. This is a factor of 5 greater
than the energy scaleE determined from the onset of
bifur-cation. Additional measurements of resonator detectors sug-gest E and Econd are comparable for a variety of inductor
volumes and critical temperatures.39,40 While Econd and E
are in reasonably good agreement, caution must be exercised when comparing these quantities, because knowledge of the absolute power level at the resonator is difficult to ascertain. This uncertainty arises from the changing electrical attenua-tion of the microwave coaxial cable upon cooling and, in particular, impedance mismatch between the 50 X coaxial transmission line and the on-chip coplanar waveguide. Additionally, the superconducting gap D is current dependent and deviates from the zero current value near bifurcation.41
III. OPTICAL RESPONSE AND NOISE EQUIVALENT POWER
During detection, incident energy absorbed by the detec-tor breaks Cooper pairs creating quasiparticles. As predicted by the Mattis-Bardeen theory,42this increases the dissipation dQ1i and kinetic inductance of the superconducting film. The resulting behavior in the linear regime can be seen in Figs.5(a)
and5(b). The increased dissipation and inductance decrease the resonance depth and frequency, respectively. Increasing, the readout power results in the onset of nonlinear behavior as shown in Figs. 5(c) and5(d). The complex response in S21 becomes asymmetric about resonance (/¼ 0), diminishing for generator frequencies above the resonance frequency (x > 0). The reduction can be understood in terms of reactive feedback. Increased optical illumination augments the kinetic inductance, shifting the resonance toward lower frequencies. The generator tone, set to a fixed frequency, is then situated further out of the shifted resonance reducing the resonator cur-rent. The nonlinear kinetic inductance is consequently reduced causing the resonance to move back to higher frequencies. This process continues until a stable equilibrium is achieved. Forx < 0, the feedback produces the opposite effect resulting in an augmented response.
In order to probe the resonance above bifurcation, the two branches of the response can be accessed experimentally by smoothly sweeping the generator frequency in either the upward or downward sense. As indicated in Fig. 1(b), we have accomplished bidirectional frequency sweeping using a voltage controlled oscillator for the signal generator. Small voltage steps and low-pass filtering ensured smooth fre-quency sweeping. The measured transfer function above bifurcation can be seen in Figs.6(a)and6(b). Due to the run-away positive feedback described above, most of the reso-nance circle in the complex plane is inaccessible while upward frequency sweeping. In contrast, nearly the entire upper half of the resonance circle (/ > 0) is accessible dur-ing downward frequency sweepdur-ing above bifurcation.
Usually in the linear regime, changes in the reactance produce a shift in the resonance frequency but do not affect
the resonance depth. Similarly, dissipation perturbations only change the resonance depth. In contrast, in the nonlinear regime dissipation perturbations can produce a frequency response. As discussed, on the high frequency side of the res-onance increased optical loading increases the kinetic induct-ance while reactive feedback tends to stabilize the resoninduct-ance against shifting toward lower frequencies. The additional loading, however, also increases the dissipation. The reso-nance depth and resonator current decrease, thus shifting the resonance toward higher frequencies. This effect becomes increasingly important as the readout power is increased. At sufficient resonator currents the dissipative frequency response can dominate. As shown in Figs.6(c)and6(d), the resonance in this case will instead move to higher frequen-cies with increasing optical loading producing a reversal in chirality in the complex response plane.
In order to determine the expected optical response and noise in the nonlinear regime, we have calculated the first-order perturbation to the power-shifted generator detuning dx to changes in the low-power resonance frequency xr;0 and
dissipation dQ1i using Eq.(11). This results in the expression
dx¼ dx0þ @ ~E @Q1 i dQ1 i 1@ ~E @x ; (14)
FIG. 5. Measured LEKID response below bifurcation to increasing the opti-cal illumination from 6.4 pW (solid, blue) to 7 pW (dashed, red). (a) In the linear case (a¼ .01), the resonance shifts to lower frequencies and becomes shallower. (b) Response in theS21plane. The resonance angle / shown in
this plot is defined such that on resonance /¼ 0 and is positive for genera-tor frequencies greater than xr(i.e., /¼ 0 for x ¼ 0 and / > 0 for x > 0).
Arrows indicate the measured displacement at a fixed generator frequency. In the linear case, the increased optical loading results in a symmetrical clockwise motion about resonance /¼ 0. (c) As the readout power is increased into the nonlinear regime (a¼ 0.5), the resonance is compressed toward lower frequencies. As explained in the main text, this distortion is understood in terms of reactive feedback. (d) In the complexS21plane, the
feedback causes an augmented response for / < 0 and diminished response for / > 0.
where ~E¼ E=E. The derivatives in Eq. (14)can be
calcu-lated from Eq.(10). The results are @ ~E @x ¼ 1 1þ xþ 8Q2 rx 1þ 4Q2 rx2 ~ E (15) and @ ~E @Q1i ¼ 2QrE~ 1þ 4Q2 rx2 : (16)
Note that these results are only valid for slow variations of xr;0 and dQ1i well below the adiabatic cutoff frequency
xr=2Qr. At higher frequencies, the ring-down response of the
resonator and the feedback must be considered.11This is not a limitation in practice as current instruments utilizing low-temperature detectors are normally concerned with measure-ment signals well below the adiabatic cutoff frequency and use low pass filtering to eliminate higher frequency noise.
In order to apply Eq. (14), appropriate values for dx0
and dQ1i must be provided. The measured optical response in the low-power linear regime is shown in Fig. 7(a). A change in optical illumination fromPopt¼ 6:4 to 7 pW can
be seen to perturb both the resonance frequency and dissipa-tion, with values dx0¼ 8 107 and dQ1i ¼ 8 10
8,
respectively. For comparison, the measured thermal response is shown in Fig. 7(b). For both the optical and thermal response, dQ1i is plotted as a function dx0in Fig.7(c). The
similarity of the two curves indicates that the device response is independent of the source of excess quasiparticles. The rela-tive frequency-to-dissipation response is found to have a ratio dx0=dQ1i 10. Applying these results to Eq. (14), the
expected response for our device is given in Fig.8along with the corresponding measurement result. The response was obtained both on resonance (/¼ 0) and for detuning up to /¼ 6150. As previously mentioned, at sufficient resonator currents the dissipative frequency response to increased opti-cal loading can result in the resonance shifting tohigher fre-quencies. The crossover to this behavior is indicated by a red contour where dx¼ 0. The fractional error between the theo-retical prediction and measurement is shown in Fig.9.
In order to calculate the expected device noise, both two-level system and amplifier contributions must be consid-ered. The fractional-frequency noise of the device at low power, given by the square root of the measured power-spectral density ffiffiffiffiffiffiSxx
p
of the fractional frequency noise dxðtÞ, is shown in Fig. 7(d). From this, a value of dx0¼ 1
1081=Hz1=2at 10 Hz can be used in Eq.(14)to calculate
the expected frequency noise in the nonlinear regime. Additionally, we assume that this value of dx0 is suppressed
as P1=4diss as previously observed by Gao et al.26,27 The TLS fluctuations in the capacitor dielectric which produce this frequency noise have not been observed to produce dissipa-tion fluctuadissipa-tions.45 Thus, for the TLS noise, dQ1i ¼ 0 and no dissipative frequency response is possible. For the amplifier contribution, we have assumed a Tn¼ 6 K noise
temperature of our cryogenic amplifier. The fluctuations in
FIG. 6. Measured LEKID response in the bifurcation regime to a change in optical loading from from 6.4 pW (solid, blue) to 7 pW (dashed, red). (a) Above bifurcation (a¼ 3), feedback results in a reduction in the frequency shift. As indicated by the arrows, the upper curves were taken while upward sweeping while the lower curves were taken while downward sweeping. (b) As a result of reactive feedback, the response inS21is considerably
dimin-ished but maintains a clockwise rotation. Here, only the downward fre-quency sweep is shown. (c) At sufficient readout power, the reduction in the current due to the increased dissipationQ1i causes the resonance to shift to
greater frequency upon an increase in optical loading. Notice that at some generator detuning, there is in fact no frequency response. (d) The resulting motion inS21in this case reverses sense to a counterclockwise rotation.
FIG. 7. Measured response and noise in the linear regime. (a) Fractional fre-quency shift dx (þ) and dissipation shift dQ1
i (o) as a function of blackbody
illumination. (b) Fractional frequency shift dx (þ) and dissipation shift dQ1 i
(o) as a function of mixing chamber temperature. The initial rise in dx at low temperature can be understood from TLS effects. A fit of dx to the Mattis-Bardeen theory42including a TLS contribution21is shown (solid line) along with the corresponding prediction for dQ1
i (dashed line). The discrepancy
between the measured data and theory has been observed in numerous TiN and NbTiN devices and is an active area of research.43,44(c) Dissipation shift dQ1
i versus fractional frequency shift dx. Here, data taken by adjusting
the blackbody illumination are marked with a dot (.) and data taken by adjusting the mixing chamber temperature is indicated with an x. (d) Fractional frequency noiseS1=2
xx measured near resonance in the linear
S21¼ ð4kTn=PgÞ1=2are then converted to dissipation and
fre-quency fluctuations. Both the calculated TLS and amplifier frequency noise contributions are shown in Fig.10. These can be summed in quadrature to yield the total device frequency noise. Both the calculated dissipation and total frequency noise are shown alongside the measured results in Fig.11.
Combining the response in Fig. 8 with the noise in Fig. 11 yields the device noise-equivalent power (NEP) in the dissipation and frequency quadratures shown in Fig. 12. These independent quadratures may be combined, accord-ing to NEP2¼ NEP2dissþ NEP
2
f req. The best NEP of 2
1016W=Hz1=2 at 10 Hz was obtained at the highest
FIG. 8. LEKID response to a change in optical loading from 6.4 to 7 pW. (a) Calculated dissipation response dQ1i . (b) Calculated frequency response dx
obtained using Eq.(14). (c) Measured dissipation response. (d) Measured frequency response. The increased optical illumination produces excess qua-siparticles. As can be seen in ((a) and (c)), this causes a uniform increase in the resonator dissipation dQ1i independent of the readout power or
genera-tor detuning angle /. The increasednqpalso produces an additional
react-ance which causes a low-power frequency shift dx0. At higher powers, the
observed frequency response depends on both dQ1
i and dx0 as well as a
feedback termð1 @ ~E
@xÞ, according to Eq.(14). Shown in ((b) and (d)), above
logðaÞ > 0:2 is the bifurcation regime where a large portion of the reso-nance is inaccessible (/ < 0) and the frequency response is suppressed by the feedback. The red contours indicates dx¼ 0. Above this contour and at small, positive /, the frequency response contributed by dQ1i dominates
and dx < 0, indicating the resonance has moved to higher frequencies under the increased illumination. Note that for the measurement, an automatic data taking procedure utilized a fixed frequency step for xg while downward
sweeping. The steep dependence of / on xgnear resonance resulted in the
region at small positive / above bifurcation to not be accessed. Future meas-urements can decrease the frequency step for xgwhen approaching xr to
obtain small, fixed steps in /.
FIG. 9. Comparison of calculated and measured response. The fractional error between the calculated and measured response is computed using the data shown in Fig. 7 and the equation jmeasured response—calculated responsej/measured response. Shown are the fractional error in (a) the dissipa-tion response dQ1
i and (b) the fractional frequency response dx. For dx, the
stripe of large errors in the region logðaÞ > 0:5 is the result of division by a diminishing measured dx. Summing over all the data shown in (a), the RMS fractional error for the dissipation response is 0.20. For (b), the RMS error is 0.22 excluding points where the measured dx approaches 0.
FIG. 10. Calculated contributions to the measured LEKID fractional fre-quency noiseS1=2
xx. (a) TLS contribution. Here, we have assumed that the
TLS noise is suppressed by frequency feedback but not dissipative feedback in the nonlinear regime. Additionally, we have assumed the TLS noise fluc-tuations are suppressed asP1=4dissas previously observed by Gaoet al.26,27(b) Amplifier contribution assuming a 6 K noise temperature. The TLS contribu-tion dominates throughout nearly the entire parameter space.
FIG. 11. Device noise at 10 Hz. (a) Calculated dissipation noise. (b) Calculated frequency noise obtained using Eq.(14). (c) Measured dissipa-tion noise. (d) Measured frequency noise. For ((a) and (c)), the observed dis-sipation noise improvement with increasinga is due to the straight-forward improvement in signal-to-noise resulting from using an increased generator power Pg relative to the fixed amplifier noise temperatureTn¼ 6 K. The
RMS fractional error comparing (a) and (c) is 0.84. For ((b) and (d)), the improvement in the frequency noise with increasinga results from a combi-nation of an increasingPgrelative to the amplifier noise, a decrease in the
TLS noise with stored resonator energy, and the frequency feedback term in the denominator of Eq.(14). The frequency feedback above logðaÞ > 0:2 is maximum around /¼ 45 and diminishes for smaller / resulting in
increased dx fluctuations near /¼ 0. The RMS fractional error comparing (b) and (d) is 0.75. In order to determine the measured noise, a time stream of fractional frequency perturbations dxðtÞ and dissipation perturbations dQ1i ðtÞ were taken at a variety of detuning angles / and values of the nonli-nearity parametera. The square root of the measured power-spectral density was then obtained, yielding the frequency noise ffiffiffiffiffiffiSxx
p
and dissipation noise ffiffiffiffiffiffiffiffi
SQQ
p
, respectively. As noted in the caption of Fig.8, the use of fixed fre-quency steps for xg rather than small, fixed steps in / while downward
sweeping resulted in the measurement not accessing the region at small posi-tive / above bifurcation.
readout power at a detuning angle of /¼ 40and is shown in Fig.13. Note that this is a10 fold improvement over the best NEP below bifurcation. We emphasize that this gain is the result of two mechanisms. First, as previously noted, increas-ing the readout power decreases the effects of TLS noise while also overcoming amplifier noise. This straightforward increase
in signal-to-noise substantially explains the NEP improve-ment. However, this is not the whole story. Equation(14) pro-vides an additional mechanism for improving NEPfreq. Due to the dissipative dQ1i term in this equation, changes in the qua-siparticle density from the optical signal result in both a reac-tive and dissipative frequency response. At high powers and near resonance, the dissipative frequency response dominates. However, as the TLS noise has no dissipative contribution, it is simply suppressed by the frequency feedback termð1 @ ~E
@xÞ.
The difference in the behavior of the frequency response and noise above the onset of bifurcation produces a region at high powers and near resonance with a substantially improved NEPfreq.
The best measured NEP is a factor of two above photon-noise limited performance for the current optical illumina-tion. In order to achieve photon-noise limited operation, a number of optimizations can be made. First, as indicated in the inset of Fig. 13, the simulated dual-polarization optical efficiency of this device under the experimental conditions was gopt 0:3. By including an anti-reflection coating, backshort, and tuning the TiN sheet resistance, we find that the optical efficiency can be improved to gopt 0:6. Implementing these changes would then give a modest 1.4 improvement in the NEP. Also, the fractional frequency noise S1=2
xx has been observed to decrease linearly with
increased temperature. Thus, operating at modestly increased temperatures, while taking care that the thermally generated quasiparticles remain negligible compared with those that are optically generated under expected loading conditions, would provide an improved NEP. Implementing these changes would potentially allow the current device to oper-ate with photon-noise limited performance under the typical illumination conditions found in ground-based, far-infrared astronomy.
IV. CONCLUSION
We have characterized the behavior of a lumped-element kinetic inductance detector optimized for the detec-tion of far-infrared radiadetec-tion in the linear and nonlinear regime. The device was fabricated from titanium nitride, a promising material due to its tunableTc, high intrinsic qual-ity factor, and large normal state resistivqual-ity. The measure-ments were performed under 6.4 pW of loading which is comparable to or somewhat less than the expected loading for ground based astronomical observations. The device was driven nonlinear by a large readout power which is desirable due to the suppression of two-level system noise in the ca-pacitor of the device at high power and the diminishing im-portance of amplifier noise at large signal powers. At sufficient readout powers, the transfer function of the detec-tor bifurcates. By smoothly downward frequency sweeping a voltage controlled oscillator, we were able to access the upper frequency side of the resonance. The best noise equiv-alent power in this regime was of 2 1016W=Hz1=2 at
10 Hz, a 10 fold improvement over the sensitivity below bifurcation.
Two practical conclusions can be drawn from this work. First, the onset of bifurcation can be increased simply by
FIG. 12. Calculated and measured NEP at 10 Hz under 6.4 pW of optical loading. (a) Calculated dissipation NEP. (b) Calculated frequency NEP. (c) Measured dissipation NEP. (d) Measured frequency NEP. Note in both fre-quency NEP subplots there is a band in the bifurcation region (logðaÞ >0:2Þ which exhibits a dramatically increasing NEP. As shown in Fig.8, in this region there is a vanishing frequency response (dx¼ 0) resulting in diminished device performance. In contrast, at high powers and near reso-nance, the dissipative frequency response dominates. As the TLS noise has no dissipative contribution, it is simply suppressed by frequency feedback. This results in a region with a significantly improved NEPfreq.
FIG. 13. Best achieved device noise equivalent power calculated using NEP2¼ NEP2
dissþ NEP 2
f req. This NEP was measured deep in the
bifurca-tion regime at log(a)¼ 1.7 and on the high frequency side of the resonance at /¼ 40. Close inspection of Fig.10reveals that under these conditions,
the TLS noise contribution to the frequency noise is suppressed below the amplifier contribution. Thus, the NEP shown here is limited by uncorrelated amplifier noise in both quadratures. The dashed line indicates the expected photon-noise limited NEPphoton¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2P0hð1 þ n0Þ
p
, whereP0is the optical
illumination and n0 is the occupation number. Inset. Simulated optical
absorption with the current measurement setup (solid, gopt 0:3) and after
decreasingQr. This allows operation at high readout powers without necessitating the use of a smooth, downward fre-quency sweep. While this provides a mechanism for achiev-ing improved device performance, the decreasedQr results in each pixel occupying a larger portion of frequency space. This proportionally decreases the multiplexing factor of the array resulting in increased electronics costs. Second, if the TLS noise of the device can be engineered below the ampli-fier noise contribution, increased pixel performance can be achieved by operating just below bifurcation on the low-frequency side of the resonance. As previously observed16 and shown here, the optical frequency response is enhanced in this region. Meanwhile, the amplifier noise is unaffected by the resonator nonlinearity. The NEPfreq is consequently improved.
The results presented are of general interest to the low-temperature detector community focusing on microresonator detectors. First, the included nonlinear resonator fitting model allows extraction of the useful resonator parameters at large readout powers when the kinetic inductance is the dom-inant device nonlinearity. Next, while increasing the readout complexity, the technique of smooth downward frequency sweeping can significantly increase the detector performance compared to operation below the onset of bifurcation while maintaining a high resonator Qr necessary for achieving dense frequency multiplexing. Note that this technique can be simultaneously applied to all resonator in an imaging array, shifting the resonances uniformly and preserving the pixel frequency spacing. Finally, the observation that the scaling energy E is of order the inductor condensation
energy allows a useful estimate of the onset of nonlinear behavior and hysteresis. We expect that a variety of experi-ments, particularly kinetic-inductance based detectors for sub-mm astronomy and dark-matter detection, will benefit from this work.
ACKNOWLEDGMENTS
The authors wish to thank Teun Klapwijk and David Moore for useful discussions relating to this work. This work was supported in part by the Keck Institute for Space Science, the Gordon and Betty Moore Foundation. Part of this research was carried out at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The devices used in this work were fabricated at the JPL Microdevices Laboratory. L. Swenson acknowledges the support from the NASA Postdoctoral Program. L. Swenson and C. McKenney acknowledge funding from the Keck Institute for Space Science. #2012. All rights reserved.
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